Convergence of power series
Some thoughts on the convergence of power series – with pictures.
Consider the three functions
f1(x) = 1 / (1 – x2)
f2(x) = 1 / (1 + x2)
f3(x) = √(1 + x2)
If we expand each of these in a power series around x = 0 we find that
f1(x) = Σ x2n
f2(x) = Σ (-1)nx2n
f3(x) = Σ x2n Binomial(1/2,n)
and that each converges for -1 < x < 1.
In the case of f1, it's easy to see why that's the case. Here's the graph of f1, and it behaves badly at -1 and 1.
In the case of f2, it's not immediately obvious why reaching -1 and 1 causes problems. Here's the graph of f2, and it certainly doesn't behave badly at all at these points.
On the other hand, the graph of what we get by summing the first few terms in the power series for f2 looks like this, which does behave badly at -1 and 1.
This is easy to explain. If we look at |f2| as a function of a complex variable, we see that f2 has poles at i and –i, and the location of those poles limits the radius of convergence of the power series.
In the case of f3, it's slightly more complicated. Here's what f3 looks like. It also doesn't behave badly at -1 or 1.
Here's what |f3| looks like as a function of a complex variable, and there aren't any poles there to cause problems. What's going on here?
What's causing problems in this case are the branch cuts that you need to define for f3. Here's a graph of the imaginary part of f3. Note that it's the locations of the branch cuts that limit the radius of convergence of the power series.